Rule of Commutation

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Theorem

Conjunction

Conjunction is commutative:

Formulation 1

$p \land q \dashv \vdash q \land p$

Formulation 2

$\vdash \paren {p \land q} \iff \paren {q \land p}$


Disjunction

Disjunction is commutative:

Formulation 1

$p \lor q \dashv \vdash q \lor p$

Formulation 2

$\vdash \paren {p \lor q} \iff \paren {q \lor p}$


Its abbreviation in a tableau proof is $\text{Comm}$.


Also known as

The rule of commutation is also known as the commutative law.

Note that this term is also used throughout mathematics in the context of addition and multiplication of numbers:

the Commutative Law of Addition
the Commutative Law of Multiplication

so it is wise to be aware of the context.


Also see


Technical Note

When invoking the Rule of Commutation in a tableau proof, use the {{Commutation}} template:

{{Commutation|line|pool|statement|depends|type}}

where:

line is the number of the line on the tableau proof where Rule of Commutation is to be invoked
pool is the pool of assumptions (comma-separated list)
statement is the statement of logic that is to be displayed in the Formula column, without the $ ... $ delimiters
depends is the line (or lines) of the tableau proof upon which this line directly depends
type is the type of Rule of Commutation: specifically Disjunction or Conjunction, whose link will be displayed in the Notes column.


Sources